where \(y^{(6)}\)  denotes the sixth-order derivative of \(y\) .
However, on substituting a periodic solution \(y(x)={\rm e}^{{\rm i}k x}\)  into this equation, it is found that the resulting expression for \(1-\lambda \)  agrees with the first three terms in (REF ), but not the fourth term. This means that this reduced model has not taken care of the response of the foundation in a self-consistent manner. An earlier attempt at refining the leading order model (REF ) was made by [1]}. However, as was shown by [2]}, their model is not self-consistent either.
The main purpose of the current study is to forsake the Winkler foundation assumption and allow the interface to transmit both vertical and shear tractions so that the resulting refined model can correctly predict the entire right hand side of (REF ). To this end, we derive a Timoshenko-type plate theory for a prestressed plate using two different approaches, a standard expansion method as used by [2]} and the expansion scheme developed in [4]} where a consistent plate theory was proposed for compressible hyperelastic materials. The latter methodology has been applied to derive consistent plate models for incompressible materials [5]}, growing solids [6]}, and nematic liquid crystal elastomers [7]}. Recently, [8]} have shown that the consistent plate model in [4]} can recover most existing plate equations by assuming appropriate order relations between the applied load and the aspect ratio. In deriving the above-mentioned Timoshenko-type plate theory for a pre-stressed plate, we naturally obtain an additional connection between the shear traction and the two displacement components at the interface, which then enables us to apply traction and displacement continuity exactly. Thus, our reduced model consists of two differential equations for the displacement components at the interface.
